Table A7.

Further comparison of summarized data methods with correlated variants.

	α	Allele score using individual‐level data	Allele score using summarized data	Weighted generalized linear regression	Likelihood‐based method
Positive causal effect: β X=0.2
β U=+1	0.05	0.225 (0.112) [52.8]	0.225 (0.126) [43.7]	0.280 (0.117) [66.9]	0.308 (0.128) [69.3]
	0.10	0.208 (0.058) [89.2]	0.208 (0.065) [86.5]	0.224 (0.063) [93.0]	0.228 (0.064) [92.9]
	0.20	0.201 (0.030) [99.9]	0.201 (0.033) [99.9]	0.206 (0.032) [100.0]	0.200 (0.032) [99.9]
β U=−1	0.05	0.175 (0.113) [33.8]	0.175 (0.104) [41.0]	0.122 (0.097) [26.9]	0.146 (0.108) [30.5]
	0.10	0.192 (0.058) [89.3]	0.192 (0.054) [90.7]	0.177 (0.052) [88.4]	0.184 (0.054) [92.0]
	0.20	0.198 (0.030) [100.0]	0.199 (0.027) [100.0]	0.195 (0.026) [100.0]	0.184 (0.026) [100.0]
Null causal effect: β X=0
β U=+1	0.05	0.024 (0.112) [6.1]	0.024 (0.114) [5.3]	0.079 (0.106) [11.6]	0.090 (0.115) [13.4]
	0.10	0.006 (0.059) [5.4]	0.006 (0.059) [4.9]	0.023 (0.056) [5.3]	0.023 (0.058) [6.5]
	0.20	0.002 (0.030) [5.1]	0.002 (0.030) [4.9]	0.006 (0.029) [5.3]	0.005 (0.029) [5.1]
β U=−1	0.05	−0.027 (0.112) [6.6]	−0.027 (0.113) [5.9]	−0.080 (0.106) [12.3]	−0.077 (0.113) [11.5]
	0.10	−0.007 (0.058) [5.4]	−0.007 (0.059) [5.1]	−0.023 (0.056) [7.0]	−0.019 (0.057) [5.9]
	0.20	−0.001 (0.030) [5.3]	0.001 (0.030) [5.1]	−0.006 (0.029) [5.6]	−0.005 (0.029) [4.8]

Median estimates over 10000 simulations of β _X=0.2 or β _X=0 (median standard errors) [power (%) based on nominal 95% confidence interval] from simulation study with 15 correlated instrumental variables (IVs) varying direction of confounding (β _U) and average strength of IV (α) using allele score method calculated from individual‐level data and allele score, weighted generalized linear regression and likelihood‐based methods all calculated from summarized data, with crude weights.